Consecutive integers in the form $ a^x+y^b $

Abstract

<abstract><p>Let $ a, b $ and $ k $ be integers greater than $ 1 $. For a tuple of $ k $ consecutive integers sorted in ascending order, denoted by $ T_k $, call $ T_k $ a nice $ k $-tuple if each integer of $ T_k $ is a sum of two powers of the form $ a^x+y^b $ and a perfect $ k $-tuple if each integer of $ T_k $ is a sum of two perfect powers of the form $ a^x+y^b $, respectively. Let $ N_k(a, b) $ be the number of nice $ k $-tuples and $ \widetilde{N}_k(a, b) $ be the number of perfect $ k $-tuples. For a given $ (a, b) $, it is quite interesting to find out $ N_k(a, b) $ and $ \widetilde{N}_k(a, b) $. In 2020, Lin and Cheng obtained the formula for $ N_k(2, 2) $. The main goal of this paper is to establish the formulas for $ N_k(a, b) $ and $ \widetilde{N}_k(a, b) $. Actually, by using the method of modulo coverage together with some elementary techniques, the formulas for $ \widetilde{N}_k(2, 2) $, $ \widetilde{N}_k(3, 2) $ and $ N_k(3, 2) $ are derived.</p></abstract>